$\mathbb{C}^N$-Smorodinsky-Winternitz system in a constant magnetic field

TL;DR

This paper introduces a superintegrable extension of the Smorodinsky-Winternitz system in complex N-dimensional space with a magnetic field, revealing additional constants of motion and symmetries, and connects it to the MICZ-Kepler problem.

Contribution

It generalizes the Smorodinsky-Winternitz system to complex space with magnetic field and analyzes its symmetries and algebraic structure, including a transformation to the MICZ-Kepler problem.

Findings

The system has 2N Liouville integrals and additional constants of motion.

The symmetry algebra of the system is explicitly computed.

The magnetic field does not qualitatively alter the system's properties.

Abstract

We propose the superintegrable generalization of Smorodinsky-Winternitz system on the $N$-dimensional complex Euclidian space which is specified by the presence of constant magnetic field. We find out that in addition to $2N$ Liouville integrals the system has additional functionally independent constants of motion, and compute their symmetry algebra. We perform the Kustaanheimo-Stiefel transformation of $\mathbb{C}^2$- Smorodinsky-Winternitz system to the (three-dimensional) generalized MICZ-Kepler problem and find the symmetry algebra of the latter one. We observe that constant magnetic field appearing in the initial system has no qualitative impact on the resulting system.